For a multiplicative cohomology theory E, complex orientations are in bijective correspondence with multiplicative natural transformations to E from complex bordism cohomology MU . If E is represented by a spectrum with a highly structured multiplication, we give an iterative process for lifting an orientation MU → E to a map respecting this extra structure, based on work of Arone–Lesh. The spa...

In this note we describe aspects of the cohomology of coherent sheaves on a complete toric variety X over a field k and, more generally, the local cohomology, with supports in a monomial ideal, of a finitely generated module over a polynomial ring S. This leads to an efficient way of computing such cohomology, for which we give explicit algorithms. The problem is finiteness. The i local cohomol...

This article is based on five lectures the author gave during the summer school, Interactions between Homotopy Theory and Algebra, from July 26–August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Gr...

Local cohomology was discovered in the 1960s as a tool to study sheaves and their cohomology in algebraic geometry, but have since seen wide use in commutative algebra. An example of their use is to answer the question: how many elements are necessary to generate a given ideal, up to radical? For example, consider two planes in 4-space meeting at a point. The vanishing ideal I = (x, y) ∩ (u, v)...

We present a closed model structure for the category of pro-spectra in which the weak equivalences are detected by stable homotopy pro-groups. With some bounded-below assumptions, weak equivalences are also detected by cohomology as in the classical Whitehead theorem for spectra. We establish an Atiyah-Hirzebruch spectral sequence in this context, which makes possible the computation of topolog...